Indirect Maximizing

PFStudent

Homework Statement

Given,
$${{(x - 7)}^{2}} + {{(y - 3)}^{2}} = {{8}^{2}}$$

What is,
$$max(3x+4y)$$

None really.

The Attempt at a Solution

Letting,
$$3x+4y = C$$

When I get to the point where I have,
$${y} = {{\frac{-3x}{4}}+{\frac{C}{4}}}$$

Then substitute that in to,
$${{(x-7)}^{2}} + {{(y-3)}^{2}} = {{8}^{2}}$$

I get,
$${{\left(x - 7\right)}^{2}} + {{\left({\left({{\frac { - 3x}{4}} + {\frac {C}{4}}}\right)} - 3\right)}^{2}} = {{8}^{2}}$$

However, I am not sure how to proceed from here since I have two unknowns: $x$ and $$C$$.

So, how do I proceed from here?

Thanks,

-PFStudent

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rootX
Can you lagrange?
If I am thinking right, then this is asking you the max on
z = 3x+4y when this intersects (x-7)^2 .. equation

rootX
If you go by your way
I would suggest to differentiate the final equation with respect to C (you are maximizing C)
And you will find y = Ax+C equation (with some numbers)
And now, you know it should agree with your (x-7)^2+(y-.. equation
This should work.

Use Lagrange if you know it. It's lot faster and easier

PFStudent
Hey,

If you go by your way
I would suggest to differentiate the final equation with respect to C (you are maximizing C)
And you will find y = Ax+C equation (with some numbers)
And now, you know it should agree with your (x-7)^2+(y-.. equation
This should work.

Use Lagrange if you know it. It's lot faster and easier

Well the way I am interpreting this problem is that in the equation,

$$max(3x+4y) = max(C)$$

Where $$max(C)$$ is a constant.

Additionally, if I differentiate as follows,

$${\frac{d}{dC}{\left[}}{{\left(x - 7\right)}^{2}} + {{\left({\left({{\frac { - 3x}{4}} + {\frac {C}{4}}}\right)} - 3\right)}^{2}}{\right]} = {\frac{d}{dC}{\left[}}{{8}^{2}}{\right]}$$

How am I supposed to differentiate: implicitly or partially with respect to $$C$$?.

In addition, taking the derivative and setting it equal to zero will only yield the values that maximize the original function--however I still do not see how this will find, max(3x+4y).

Thanks,

-PFStudent

rootX
partially: treat x as constant.
so you will get C = something*x+some numbers
now substitute C in 3x+4y = C equation
and you will be some line
So, now find intersection of this line with original function(would give u max/min)

I think max(3x+4y) means you take x and y value from your function domain. So, finding function max when x and y are in 3x+4y relationship should give u the answer...or something like that

rootX
my interpretation:

Draw a cylinder in x-y-z co-od with that is defined by (x-7)^2 .. equation when z = 0
Draw a plane define by z=3x+4y

you will get a slanted disk, and they are asking for max of that disk

ice109
parameterize the constraint then plug that parameterization into your function. then maximize subject to the parameterization. the easieast way to parameterize your constraints is x = f(y) then you'll have to check max's on two parameterizations. if you still can't get it i'll post more.

actually just use lagrange multipliers.

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